Pawlak粗糙集的若干重要性质

A Few Important Properties of Pawlak Rough Sets

为了更深入地了解任意论域(不仅仅是有限论域)上的粗糙集理论,首先从任意论域上的等价关系R出发,提出了R粗糙集和R精细集的定义,该定义与集合的上、下近似无关。在此基础上研究了R精细集的性质,提出并证明了任意论域上R精细集的判定定理和运算封闭性定理。然后,讨论了上、下R近似的性质,提出并证明了上、下R近似的表示定理、比较定理和拓扑结构定理。最后研究了知识库的相关性,给出了正域表示定理和知识库相关性判定定理。这些结果在一定程度上丰富了Pawlak粗糙集理论。

In order to gain a deeper comprehension of rough set theory in arbitrary universes （ not only in finite universes）, definitions of R - rough set and R - accurate set are presented from the equivalence relations R in arbitrary universes. These definitions are independent of the upper and lower approximations. Based on this, the properties of R - accurate sets are investigated, a determination theorem and a closeness theorem of operations of accurate sets are put forward and proved. Then the properties of the upper and lower approximations are discussed, again, a representation theorem, a comparison theorem and a topological structure theorem of the upper and lower approximations of rough sets are presented and proved. Finally the dependency of a knowledgebase is researched ; a representation theorem of positive areas and a determination theorem of dependency of the knowledgebase are given. These results enrich to a certain extent Pawlak＇s rough set theory. The conception of accurate set in Chinese is updated.